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Expressions and Equations

July 17, 2025
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Shivohm Karogal

3

Min Read

AI Summary

These notes cover understanding and interpreting linear equations both graphically and algebraically, including how to graph proportional relationships and calculate slope. They also explain solving single-variable linear equations by isolating variables and classifying solutions, as well as solving systems of two linear equations through graphing and algebraic methods like substitution and elimination.

Objective

Understand and interpret both linear and graphical equations.

 

Core Concept

Mastering equations is essential for solving more complex algebraic problems and analyzing relationships between variables.


Graph Proportional Relationships and Interpret Slope

  1. Create a Table:

Miles (X)

Money Earned (Y)

1

10

  1. Plot Points on a Graph: Plot each coordinate pair to visualize the relationship.

  2. Calculate the Slope: Use the formula:Slope = Rise / Run

  3. Find the Slope Value: Count vertical change (rise) and horizontal change (run) between two points.


Derive the Linear Equation (y = mx + c)

  1. Find the Slope (m): Use rise/run from two points on the graph.

  2. Identify the Y-Intercept (c): This is where the line crosses the Y-axis.

  3. Form the Equation: Combine both: y = mx + c


Solving Single-Variable Linear Equations

  1. Simplify both sides and combine like terms.

  2. Move variables to one side and constants to the other.

  3. Divide or multiply to isolate the variable.

  4. Classify the solution:

    1. One Solution: Single correct value.

    2. No Solution: Contradictory statement (e.g., 5 = 9).

    3. Infinite Solutions: Both sides match identically.


Solving and Interpreting Systems of Two Linear Equations

  • Graphically:

    • Plot both equations on a graph.

    • The intersection point gives the solution.

  • Algebraically:

    • Use substitution or elimination to find the point where both equations are true.

Key Terms
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